Almost quantum theory: classical theories with a constraint on knowledge

What kind of theory would be appropriate for an agent living in a world that is essentially classical but where there is a fundamental restriction on how much knowledge can be acquired about the physical state of any system? Formalizing such a restriction, one can define several toy theories that are found to have a rich structure similar to that of quantum theory, including a notion of coherent superposition and entanglement. These theories are also found to have analogues of a wide variety of quantum phenomena, such as complementarity, interference, teleportation, no-cloning, and many quantum cryptographic and communication protocols. The diversity and quality of these analogies provides compelling evidence for the view that quantum states are not states of reality -- as most interpretations suggest -- but rather states of knowledge that are incomplete (and cannot be completed). The question ``what is the nature of the reality to which this knowledge refers?" remains open in this research program but the phenomenon of contextuality (a consequence of the Bell-Kochen-Specker theorem) provides, I argue, our best clue for how to answer it.